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Simplifying 7a2 + -18a + 5 = 15 Reorder the terms: 5 + -18a + 7a2 = 15 Solving 5 + -18a + 7a2 = 15 Solving for variable 'a'. Reorder the terms: 5 + -15 + -18a + 7a2 = 15 + -15 Combine like terms: 5 + -15 = -10 -10 + -18a + 7a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -10 + -18a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.428571429 + -2.571428571a + a2 = 0 Move the constant term to the right: Add '1.428571429' to each side of the equation. -1.428571429 + -2.571428571a + 1.428571429 + a2 = 0 + 1.428571429 Reorder the terms: -1.428571429 + 1.428571429 + -2.571428571a + a2 = 0 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + -2.571428571a + a2 = 0 + 1.428571429 -2.571428571a + a2 = 0 + 1.428571429 Combine like terms: 0 + 1.428571429 = 1.428571429 -2.571428571a + a2 = 1.428571429 The a term is -2.571428571a. Take half its coefficient (-1.285714286). Square it (1.653061225) and add it to both sides. Add '1.653061225' to each side of the equation. -2.571428571a + 1.653061225 + a2 = 1.428571429 + 1.653061225 Reorder the terms: 1.653061225 + -2.571428571a + a2 = 1.428571429 + 1.653061225 Combine like terms: 1.428571429 + 1.653061225 = 3.081632654 1.653061225 + -2.571428571a + a2 = 3.081632654 Factor a perfect square on the left side: (a + -1.285714286)(a + -1.285714286) = 3.081632654 Calculate the square root of the right side: 1.755457961 Break this problem into two subproblems by setting (a + -1.285714286) equal to 1.755457961 and -1.755457961.Subproblem 1
a + -1.285714286 = 1.755457961 Simplifying a + -1.285714286 = 1.755457961 Reorder the terms: -1.285714286 + a = 1.755457961 Solving -1.285714286 + a = 1.755457961 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + a = 1.755457961 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + a = 1.755457961 + 1.285714286 a = 1.755457961 + 1.285714286 Combine like terms: 1.755457961 + 1.285714286 = 3.041172247 a = 3.041172247 Simplifying a = 3.041172247Subproblem 2
a + -1.285714286 = -1.755457961 Simplifying a + -1.285714286 = -1.755457961 Reorder the terms: -1.285714286 + a = -1.755457961 Solving -1.285714286 + a = -1.755457961 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + a = -1.755457961 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + a = -1.755457961 + 1.285714286 a = -1.755457961 + 1.285714286 Combine like terms: -1.755457961 + 1.285714286 = -0.469743675 a = -0.469743675 Simplifying a = -0.469743675Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.041172247, -0.469743675}
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